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Contact Logic is finitary for unification with constants
Philippe Balbiani, Cigdem Gencer
To cite this version:
Philippe Balbiani, Cigdem Gencer. Contact Logic is finitary for unification with constants. Workshop
on Logic and Cognition, Dec 2016, Canton, China. �hal-02936482�
Contact Logic is finitary for
unification with constants
Philippe Balbiani1 and C¸ i˘gdem Gencer2,3 1
Institut de recherche en informatique de Toulouse CNRS — Toulouse University
Toulouse, France
2
Faculty of Engineering and Natural Sciences Sabancı University
Istanbul, Turkey
3Faculty of Arts and Sciences
Aydın University Istanbul, Turkey
Abstract. Contact Logic is a formalism for reasoning about the rela-tion of contact between regions. We introduce a new inference problem for Contact Logic, the unification problem, which extends the validity problem by allowing one to replace variables by terms before testing for validity. Our main result is the proof that unification with constants in Contact Logic is finitary.
Keywords: Contact Logic, unification with constants, unification type.
1
Introduction
Contact Logic is a formalism for reasoning about the relation of contact be-tween regions represented by Boolean terms. Its language is obtained from the language of Boolean algebras by the addition of a binary predicate representing this relation. Its role in artificial intelligence and computer science stems from the importance of spatial information in systems for natural language understand-ing, robotic navigation, etc [8, 22]. Different variants of Contact Logic have been proposed [19–21, 24, 26]. Some of them are based on the predicate of boundedness or the predicate of connectedness. Their semantics can be given by interpreting terms and formulas either in topological spaces [9–11], or in frames [7, 25]. The above-mentioned applications of Contact Logic require the resolution of unification problems. There is a wide variety of situations where unification problems arise. Suppose the formula ϕ(p1, . . . , pm) describes a given geographic
configuration of constant regions p1, . . . , pm and the formula ψ(x1, . . . , xn)
rep-resents a desirable geographic property of variable regions x1, . . . , xn. It may
happen that ϕ(p1, . . . , pm) → ψ(x1, . . . , xn) is not valid in the considered
of terms such that ϕ(p1, . . . , pm) → ψ(a1, . . . , an) is valid in this environment.
Moreover, one may be interested to obtain, if possible, the most general n-tuples (a1, . . . , an) of terms such that ϕ(p1, . . . , pm) → ψ(a1, . . . , an) is valid.
Little is known about the unification problem in Contact Logic. For example, without the unary predicate of connectedness, elementary unification is N P -complete whereas unification with constants is decidable, its exact complexity being still unknown. See [5] for details about the computability of the unification problem in Contact Logic without the unary predicate of connectedness. A logic is said to be unitary when the solutions of a unification problem can always be represented by a most general one whereas it is said to be finitary when the solutions of a unification problem can always be represented by finitely many maximal ones. In this paper, we prove that unification with constants in Contact Logic either with, or without the unary predicate of connectedness is finitary.
2
Syntax
We adopt the standard rules for omission of the parentheses.
Atoms Let CON be a countable set of propositional constants (with typical members denoted p, q, etc) and VAR be a countable set of propositional variables (with typical members denoted x, y, etc). Let (p1, p2, . . .) be an enumeration of
CON without repetitions and (x1, x2, . . .) be an enumeration of VAR without
repetitions. A propositional atom is either a constant, or a variable. Let ATO be the set of all atoms (with typical members denoted α, β, etc).
Terms The Boolean terms (denoted a, b, etc) are defined as follows: – a ::= α | 0 | a?| (a ∪ b).
The other Boolean constructs for terms (for instance, 1 and ∩) are defined as usual. We use the following notations for terms: a0 for a? and a1for a. Reading terms as regions, the constructs 0, ? and ∪ should be regarded as the empty region, the complement operation and the union operation. As a result, the constructs 1 and ∩ should be regarded as the full region and the intersection operation. For all nonnegative integers m, n, let TERm,nbe the set of all terms
whose constants form a subset of {p1, . . . , pm} and whose variables form a subset
of {x1, . . . , xn}. Let TER be the set of all terms.
Formulas The formulas (denoted ϕ, ψ, etc) are defined as follows: – ϕ ::= C(a, b) | a ≡ b | con(a) | ⊥ | ¬ϕ | (ϕ ∨ ψ).
Here, a and b are terms whereas C is the binary predicate of contact, ≡ is the binary predicate of equality and con is the unary predicate of connectedness. For C, ≡ and con, we propose the following readings:
– C(a, b): “the regions denoted a and b are in contact”, – a ≡ b: “the regions denoted a and b are equal”, – con(a): “the region denoted a is connected”.
The other connectives for formulas (for instance, > and ∧) are defined as usual. We use the following notation for formulas: a ≤ b for a ∪ b ≡ b. A formula ϕ is equational if ≡ is the only predicate possibly occurring in ϕ. For all nonnegative integers m, n, let FORm,n be the set of all formulas whose constants form a
subset of {p1, . . . , pm} and whose variables form a subset of {x1, . . . , xn}. Let
FOR be the set of all formulas. Note that FOR is denoted Cc in [19–21] and Lc in [24].
Example 1. Within the context of the unification problem, we will interest with the following formulas:
– ϕ01 = x ≡ 0 ∨ x ≡ 1,
– ϕpq = C(p, q) → x 6≡ 0 ∧ x ≤ p ∪ q.
The first formula is equational. It says that the region denoted x is either the empty region, or the full region. As for the second formula, it says that if the regions denoted p and q are in contact then the region denoted x is a nonempty region included in the region denoted p ∪ q.
Substitutions A substitution is a function σ : VAR −→ TER which moves at most finitely many variables. Given a substitution σ, let ¯σ : TER ∪ FOR −→ TER ∪ FOR be the endomorphism such that for all variables x, ¯σ(x) = σ(x). The composition of the substitutions σ and τ is the substitution σ ◦ τ such that for all x ∈ VAR, (σ ◦ τ )(x) = ¯τ (σ(x)).
Example 2. Within the context of the unification problem, we will interest with the following substitutions:
– the substitution σ0such that σ0(x) = 0 and for all variables y, if x 6= y then
σ0(y) = y,
– the substitution σ1such that σ1(x) = 1 and for all variables y, if x 6= y then
σ1(y) = y,
– the substitution σp such that σp(x) = p ∪ (q ∩ x) and for all variables y, if
x 6= y then σp(y) = y,
– the substitution σq such that σq(x) = q ∪ (p ∩ x) and for all variables y, if
x 6= y then σq(y) = y.
3
Semantics
The semantics of Contact Logic can be given by interpreting terms and formulas either in topological spaces [7, 26], or in frames [7, 25].
Topological semantics The best way to understand the meaning of the predicates of contact and connectedness is by interpreting them in topological spaces, i.e. structures of the form (X, τ ) where X is a nonempty set (with typical members denoted A, B, etc) and τ is a set of subsets of X such that
– ∅ ∈ τ , – X ∈ τ ,
– if {Ai: i ∈ I} is a finite subset of τ thenT{Ai: i ∈ I} ∈ τ ,
– if {Ai: i ∈ I} is a subset of τ then S{Ai: i ∈ I} ∈ τ .
The subsets of X in τ are called open sets whereas their complements are called closed sets. The most important topological spaces considered within the context of Contact Logic are for all n ≥ 1, the topological space (Rn, τ
Rn), i.e. the real
space Rn together with its ordinary topology τ
Rn. See [19–21, 26] for details. Let
(X, τ ) be a topological space. Let Intτ and Clτ denote the interior operator and
the closure operator in (X, τ ). A subset A of X is regular closed if Clτ(Intτ(A)) =
A. Regular closed subsets of X will also be called regions. It is well-known that the set RC(X, τ ) of all regular closed subsets of X forms a Boolean algebra (RC(X, τ ), 0X, ?X, ∪X) where for all A, B ∈ RC(X, τ ):
– 0X = ∅,
– A?X = Cl
τ(X \ A),
– A ∪XB = A ∪ B.
As a result, for all A, B ∈ RC(X, τ ), 1X = X and A ∩XB = Clτ(Intτ(A ∩ B)).
Regions being regular closed subsets of X, two regions are in contact if they have a nonempty intersection whereas a region is connected if it cannot be represented as the union of two disjoint nonempty regions. For this reason, we associate to the predicates C and con the relations C(X,τ ) and con(X,τ ) on the set of all regular closed subsets of X as follows:
– C(X,τ )(A, B) if A ∩ B 6= ∅,
– con(X,τ )(A) if for all nonempty regular closed subsets B, B0of X, if B ∪ B0= A then B ∩ B06= ∅.
The following conditions hold for all regular closed subsets A, B of X: – if A 6= ∅ then C(X,τ )(A, A),
– if C(X,τ )(A, B) then C(X,τ )(B, A).
If τ only lets ∅ and X being open then the following condition holds for all regular closed subsets A, B of X:
– if A 6= ∅ and B 6= ∅ then C(X,τ )(A, B).
If X cannot be represented as the union of two disjoint nonempty open sets then the following conditions hold for all regular closed subsets A of X:
– if A 6= ∅ and A 6= X then C(X,τ )(A, Cl
τ(X \ A)),
A valuation on X is a map V associating with every atom α a regular closed subset V(α) of X. Given a valuation V on X, we define
– ¯V(α) = V(α), – ¯V(0) = ∅, – ¯V(a?) = Cl
τ(X \ ¯V(a)),
– ¯V(a ∪ b) = ¯V(a) ∪ ¯V(b).
Thus, every term is interpreted as a regular closed subset of X. A model on (X, τ ) is a structure M = (X, τ, V) where V is a valuation on X. The connectives ⊥, ¬ and ∨ being classically interpreted, the satisfiability of a formula ϕ in M (in symbols M |= ϕ) is defined as follows:
– M |= C(a, b) if C(X,τ )( ¯V(a), ¯V(b)),
– M |= a ≡ b if ¯V(a) = ¯V(b), – M |= con(a) if con(X,τ )( ¯V(a)).
A formula ϕ is valid in (X, τ ) if for all valuations V on X, (X, τ, V) |= ϕ. A formula ϕ is satisfiable in (X, τ ) if there exists a valuation V on X such that (X, τ, V) |= ϕ.
Galton’s perspective A relational perspective is suggested by Galton [13] who introduces the notion of adjacency space. Galton’s spaces are frames (W, R) where W is a nonempty set of cells and R is an adjacency relation between cells. Galton defines regions to be sets of cells. He also defines two regions A and B to be in contact if some point in A is adjacent to some point in B. This definition relates Galton’s adjacency spaces to the relational semantics of modal logic which makes it possible to use methods from modal logic for studying region-based theories of space. The truth is that the above-mentioned topological semantics and the relational perspective suggested by Galton are equivalent [24, 25]. In this paper, we adopt Galton’s perspective.
Relational semantics A frame is a structure of the form (W, R) where W is a nonempty set (with typical members denoted s, t, etc) and R is a binary relation on W . Let (W, R) be a frame. We will denote by R+ the transitive closure of
R, i.e. the least transitive relation on W containing R. Subsets of W will also be called regions. Naturally, two regions are in contact if they have R-related elements whereas a region is connected if there is an R-path between any pair of elements in that region. For this reason, we associate to the predicates C and con the relations C(W,R) and con(W,R) on the set of all subsets of W as follows:
– C(W,R)(A, B) if R ∩ (A × B) 6= ∅,
– con(W,R)(A) if A × A ⊆ R+.
If R is reflexive and symmetric then the following conditions hold for all subsets A, B of W :
– if C(W,R)(A, B) then C(W,R)(B, A).
If R = W × W then the following condition holds for all subsets A, B of W : – if A 6= ∅ and B 6= ∅ then C(W,R)(A, B).
If R+= W × W then the following conditions hold for all subsets A of W :
– if A 6= ∅ and A 6= W then C(W,R)(A, W \ A), – con(W,R)(W ).
A valuation on W is a map V associating with every atom α a subset V(α) of W . Given a valuation V on W , we define
– ¯V(α) = V(α), – ¯V(0) = ∅,
– ¯V(a?) = W \ ¯V(a),
– ¯V(a ∪ b) = ¯V(a) ∪ ¯V(b).
Thus, every term is interpreted as a subset of W . A model on (W, R) is a structure M = (W, R, V) where V is a valuation on W . The connectives ⊥, ¬ and ∨ being classically interpreted, the satisfiability of a formula ϕ in M (in symbols M |= ϕ) is defined as follows:
– M |= C(a, b) if C(W,R)( ¯V(a), ¯V(b)),
– M |= a ≡ b if ¯V(a) = ¯V(b), – M |= con(a) if con(W,R)( ¯V(a)).
A formula ϕ is valid in (W, R) if for all valuations V on W , (W, R, V) |= ϕ. A formula ϕ is satisfiable in (W, R) if there exists a valuation V on W such that (W, R, V) |= ϕ.
Validity and satisfiability Let C be a class of frames. We say that a formula ϕ is C-valid if for all frames (W, R) in C, ϕ is valid in (W, R). We say that a formula ϕ is C-satisfiable if there exists a frame (W, R) in C such that ϕ is satisfiable in (W, R). The C-satisfiability problem consists in determining whether a given formula is C-satisfiable. A frame (W, R) is indiscrete if R = W × W . A frame (W, R) is connected if R+ = W × W . Let C
all denote the class of all
reflexive and symmetric frames, Cinddenote the class of all reflexive, symmetric
and indiscrete frames and Ccon denote the class of all reflexive, symmetric and
connected frames. The topological counterparts of these classes of frames are the class of all topological spaces, the class of all indiscrete spaces (those spaces that only let ∅ and the full set being open) and the class of all connected spaces (those spaces that cannot be represented as the union of two disjoint nonempty open sets). For confree formulas, the satisfiability problem is known to be N P -complete in Call and Cind and P SP ACE-complete in Ccon, the situation being
generally more complex for arbitrary formulas. See [7, 19–21, 26] for details. The following formulas are Call-valid:
– C(x, y) → C(y, x).
The following formula is Cind-valid:
– x 6≡ 0 ∧ y 6≡ 0 → C(x, y).
The following formulas are Ccon-valid:
– x 6≡ 0 ∧ x 6≡ 1 → C(x, x?),
– con(1).
4
About classes of frames
Let C be a class of frames. We say that C agrees with unions if for all disjoint frames (W, R), (W0, R0) in C, there exists a frame (W00, R00) in C such that W ∪ W0 = W00 and R006⊆ IdW00. We should stress here that many natural classes of frames agree with unions. In particular,
Proposition 1. Call, Cind and Ccon agree with unions.
Proof. By the definition of what it means for classes of frames to agree with unions.
We say that C is determined by a formula ϕ if C is the class of all frames validating ϕ.
Proposition 2. Call is determined by x 6≡ 0 → C(x, x) and C(x, y) → C(y, x),
Cind is determined by x 6≡ 0 ∧ y 6≡ 0 → C(x, y) and Ccon is both determined by
x 6≡ 0 ∧ x 6≡ 1 → C(x, x?) and con(1).
Proof. Left to the reader.
We say that C is large if there exists a frame (W, R) in C such that Card(W ) ≥ 3 and R 6⊆ IdW. We should stress here that many natural classes of frames are
large. In particular, obviously, if a nonempty class of frames agrees with unions then it is large. A valuation V on a nonempty set W is balanced if for all terms a, either ¯V(a) = ∅, or ¯V(a) = W , or ¯V(a) is infinite and coinfinite. We say that C is balanced if for all formulas ϕ, if ϕ is C-satisfiable then there exists a countable frame (W, R) in C and there exists a balanced valuation V on W such that (W, R, V) |= ϕ. We should stress here that many natural classes of frames are balanced. In particular,
Proposition 3. Call, Cind and Ccon are balanced.
Proof. Let C be one of the above-mentioned classes of frames. In order to demon-strate that C is balanced, let ϕ be a C-satisfiable formula. By [7, Theorem 4.2], ϕ is satisfiable in a finite frame of C. Let (W, R) be a finite frame of C and V be a valuation on it such that (W, R, V) |= ϕ. Let W0 = W × N and R0 be the binary relation on W0 defined by (x, i)R0(y, j) if xRy. Since the frame (W, R) is finite, therefore the frame (W0, R0) is countable. Let V0 be the valuation on
(W0, R0) defined by V0(α) = V(α) × N for each atom α. As the reader may eas-ily verify by induction, ¯V0(a) = V(a) × N for each term a. It follows that the
model (W0, R0, V0) is balanced. As the reader may easily verify by induction, (W, R, V) |= ψ iff (W0, R0, V0) |= ψ for each formula ψ. Since (W, R, V) |= ϕ, therefore (W0, R0, V0) |= ϕ. Finally, it suffices now to remark that if (W, R) is in Call (respectively, in Cind, in Ccon) then (W0, R0) is in Call (respectively, in Cind,
in Ccon) too.
5
Unifiability
Now, following the standard notations and definitions about the unifiability problem in modal and description logics [2–4, 14–17, 23], we introduce the unifi-ability problem in Contact Logic. Let C be a class of frames.
Unifiable formulas We say that a substitution σ is C-equivalent to a substitution τ (in symbols σ 'C τ ) if for all variables x, σ(x) ≡ τ (x) is C-valid.
Example 3. As the reader can easily show, the substitutions σ0 and σ1
intro-duced in Example 2 are respectively C-equivalent to the substitutions σ00and σ10 such that σ00(x) = x ∩ x?, σ0
1(x) = x ∪ x? and for all variables y, if x 6= y then
σ00(y) = y and σ10(y) = y.
We say that a substitution σ is more C-general than a substitution τ (in symbols σ C τ ) if there exists a substitution υ such that σ ◦ υ 'C τ . In that case, we
also say that τ is less C-general than σ.
Example 4. As the reader can easily show, the substitutions σp and σq
intro-duced in Example 2 are respectively less C-general than the substitutions σ0pand σ0q such that σ0p(x) = x ∪ p, σ0q(x) = x ∪ q and for all variables y, if x 6= y then σ0p(y) = y and σq0(y) = y.
We say that a formula ϕ is C-unifiable if there exists a substitution σ such that ¯
σ(ϕ) is C-valid. In that case, we say that σ is a C-unifier of ϕ.
Example 5. As the reader can easily show, the substitutions σ0 and σ1
intro-duced in Example 2 are C-unifiers of the formula ϕ01 introduced in Example 1.
In other respect, the substitutions σp and σq introduced in Example 2 are
C-unifiers of the formula ϕpq introduced in Example 1.
The C-unifiability problem with constants consists in determining whether a given formula is C-unifiable. We say that a set of C-unifiers of a formula ϕ is complete if for all C-unifiers σ of ϕ, there exists a C-unifier τ of ϕ in that set such that τ C σ. An important question is: when a formula is C-unifiable, has it a minimal
Unification types We say that a C-unifiable formula ϕ is C-finitary if there exists a finite minimal complete set of C-unifiers of ϕ but there exists no with cardinality 1. We say that a C-unifiable formula ϕ is C-unitary if there exists a minimal complete set of C-unifiers of ϕ with cardinality 1. We say that unification in C is finitary if every C-unifiable formula is either C-finitary, or C-unitary and there exists a finitary formula. We say that unification in C is unitary if every C-unifiable formula is C-unitary. See [6, 12, 18] for an introduction to the unification types in logics.
6
Examples
From now on, when we write “CPL”, we mean “Classical Propositional Logic”. Let C be a class of frames.
As remarked above, the substitutions σ0 and σ1 introduced in Example 2 are
C-unifiers of the formula ϕ01introduced in Example 1. Moreover,
– σ06Cσ1 and σ16C σ0.
To see why, suppose σ0C σ1, or σ1C σ0. Without loss of generality, suppose
σ0 C σ1. Let τ be a substitution such that σ0◦ τ 'C σ1. Hence, ¯τ (σ0(x)) ≡
σ1(x) is C-valid. Since σ0(x) = 0 and σ1(x) = 1, therefore 0 ≡ 1 is C-valid: a
contradiction. In other respect,
– if C agrees with unions then σ0 and σ1 form a complete set of C-unifiers of
ϕ01.
To see why, suppose C agrees with unions and σ0and σ1do not form a complete
set of C-unifiers of ϕ01. Let τ be a C-unifier of ϕ01such that neither σ0C τ , nor
σ1C τ . Thus, neither 0 ≡ τ (x) is C-valid, nor 1 ≡ τ (x) is C-valid. Let (W, R)
and (W0, R0) be disjoint frames in C, V be a valuation on W and V0be a valuation on W0 such that neither ¯V(τ (x)) = ∅, nor ¯V0(τ (x)) = W0. Since C agrees with
unions, therefore let (W00, R00) be a frame in C such that W ∪W0= W00. Let V00be a valuation on W00such that for all atoms α, V00(α) = V(α)∪V0(α). Obviously, for all terms a, ¯V00(a) = ¯V(a) ∪ ¯V0(a). Since neither ¯V(τ (x)) = ∅, nor ¯V0(τ (x)) = W0,
therefore neither ¯V00(τ (x)) = ∅, nor ¯V00(τ (x)) = W00. Consequently, τ is not a
C-unifier of ϕ01: a contradiction. Remark that the above argument shows that if
C agrees with unions then ϕ01 is either C-unitary, or C-finitary. Finally,
– if C agrees with unions then ϕ01 is C-finitary.
To see why, suppose C agrees with unions and ϕ01 is not C-finitary. Since if C
agrees with unions then ϕ01 is either C-unitary, or C-finitary, therefore ϕ01 is
C-unitary. Let τ be a C-unifier of ϕ01 such that τ C σ0 and τ C σ1. Since σ0
and σ1 form a complete set of C-unifiers of ϕ01, therefore σ0C τ , or σ1C τ .
Without loss of generality, suppose σ0C τ . Since τ C σ1, therefore σ0Cσ1:
As remarked above, the substitutions σp and σq introduced in Example 2 are
C-unifiers of the formula ϕpqintroduced in Example 1. Moreover,
– if p 6= q then σp 6C σq and σq 6C σp.
To see why, suppose p 6= q and σpC σq, or σqC σp. Without loss of generality,
suppose σp C σq. Let τ be a substitution such that σp◦ τ 'C σq. Hence,
¯
τ (σp(x)) ≡ σq(x) is C-valid. Since σp(x) = p ∪ (q ∩ x) and σq(x) = q ∪ (p ∩ x),
therefore p ∪ (q ∩ τ (x)) ≡ q ∪ (p ∩ x) is C-valid. Thus, after replacing in p ∪ (q ∩ τ (x)) ≡ q ∪ (p ∩ x) each occurrence of q and x by 0, p ≡ 0 is C-valid: a contradiction. In other respect,
– if C agrees with unions and p 6= q then σp and σq form a complete set of
C-unifiers of ϕpq.
To see why, suppose C agrees with unions, p 6= q and σp and σq do not form
a complete set of C-unifiers of ϕpq. As a result, remark that C is large. Let τ
be a C-unifier of ϕpq such that neither σp C τ , nor σq C τ . Thus, C(p, q) →
τ (x) 6≡ 0 ∧ τ (x) ≤ p ∪ q is C-valid. Without loss of generality, we can assume that τ (x) = (p0∩ q0∩ a
00) ∪ (p0∩ q1∩ a01) ∪ (p1∩ q0∩ a10) ∪ (p1∩ q1∩ a11) for
some {p, q}-free terms a00, a01, a10, a11. Since C is large, therefore let (W, R) be
a frame in C such that Card(W ) ≥ 3 and R 6⊆ IdW. As a result, remark that
R 6= ∅. Suppose we succeed in proving the following: (1) a00≡ 0 is C-valid,
(2) a11≡ 1 is C-valid,
(3) a01∪ a10≡ 1 is C-valid,
(4) a01≤ a10 is C-valid, or a10≤ a01is C-valid.
Consequently, by (1) and (2), considered as formulas in CPL, τ (x) and (p0∩q1∩
a01) ∪ (p1∩ q0∩ a10) ∪ (p1∩ q1) are equivalent. Moreover, by (3) and (4), a10≡ 1
is C-valid, or a01 ≡ 1 is C-valid. In the former case, considered as formulas in
CPL, τ (x) and p ∪ (q ∩ a01) are equivalent. Hence, σp C τ : a contradiction.
In the latter case, considered as formulas in CPL, τ (x) and q ∪ (p ∩ a10) are
equivalent. Thus, σq C τ : a contradiction. Consequently, it remains to prove
(1), (2), (3) and (4).
(1) Suppose a00 ≡ 0 is not C-valid. Let V be a valuation on (W, R) such that
¯
V(a00) = W . Since a00is {p, q}-free, Card(W ) ≥ 3 and R 6= ∅, therefore we can
assume that V(p) ∪ V(q) 6= W and R ∩ (V(p) × V(q)) 6= ∅. Hence, (W, R, V) |= C(p, q) and ¯V(τ (x)) 6⊆ ¯V(p ∪ q): a contradiction with the C-validity of C(p, q) → τ (x) 6≡ 0∧τ (x) ≤ p∪q. Thus, a00≡ 0 is C-valid and (1) is proved. Moreover,
con-sidered as formulas in CPL, τ (x) and (p0∩q1∩a
01)∪(p1∩q0∩a10)∪(p1∩q1∩a11)
are equivalent.
(2) Suppose a11 ≡ 1 is not C-valid. Let V be a valuation on (W, R) such that
¯
V(p) = V(q) and R ∩ (V(p) × V(q)) 6= ∅. Consequently, (W, R, V) |= C(p, q) and ¯
V(τ (x)) = ∅: a contradiction with the C-validity of C(p, q) → τ (x) 6≡ 0 ∧ τ (x) ≤ p ∪ q. Hence, a11 ≡ 1 is C-valid and (2) is proved. Moreover, considered as
for-mulas in CPL, τ (x) and (p0∩ q1∩ a
01) ∪ (p1∩ q0∩ a10) ∪ (p1∩ q1) are equivalent.
(3) Suppose a01∪ a10 ≡ 1 is not C-valid. Let V be a valuation on (W, R) such
that ¯V(a01) = ∅ and ¯V(a10) = ∅. Since p 6= q, a01 and a10 are {p, q}-free and
R 6⊆ IdW, therefore we can assume that V(p)∩V(q) = ∅ and R∩(V(p)×V(q)) 6= ∅.
Thus, (W, R, V) |= C(p, q) and ¯V(τ (x)) = ∅: a contradiction with the C-validity of C(p, q) → τ (x) 6≡ 0 ∧ τ (x) ≤ p ∪ q. Consequently, a01∪ a10≡ 1 is C-valid and
(3) is proved.
(4) Suppose neither a01 ≤ a10 is C-valid, nor a10 ≤ a01 is C-valid. By
us-ing an argument similar to the one showus-ing above that σ0 and σ1 form a
complete set of unifiers of ϕ01, we obtain a frame (W00, R00) in C such that
R00 6⊆ IdW and a valuation V00 on W00 such that V¯00(a01) 6⊆ V¯00(a10) and
¯ V00(a
10) 6⊆ ¯V00(a01). Since p 6= q and a01 and a10are {p, q}-free, therefore we can
assume that V00(p) and V00(q) are such that V00(p) ⊆ ¯V00(a
01), V00(p)∩ ¯V00(a10) = ∅,
V00(q) ∩ ¯V00(a
01) = ∅, V00(q) ⊆ ¯V00(a10) and R ∩ (V00(p) × V00(q)) 6= ∅. Thus,
¯
V00(τ (x)) = ∅ and (W00, R00, V00) |= C(p, q): a contradiction with the C-validity
of C(p, q) → τ (x) 6≡ 0 ∧ τ (x) ≤ p ∪ q. Thus, a01≤ a10is C-valid, or a10≤ a01 is
C-valid and (4) is proved.
Remark that the above argument shows that if C agrees with unions and p 6= q then ϕpq is either C-unitary, or C-finitary. Finally,
– if C agrees with unions and p 6= q then ϕpq is C-finitary.
To see why, suppose C agrees with unions, p 6= q and ϕpq is not C-finitary.
Since if C agrees with unions and p 6= q then ϕpq is either unitary, or
C-finitary, therefore ϕpqis either C-unitary, or C-finitary. Since ϕpqis not C-finitary,
therefore ϕpq is C-unitary. Let τ be a C-unifier of ϕpq such that τ C σp and
τ C σq. Since σp and σq form a complete set of C-unifiers of ϕpq, therefore
σpC τ , or σq C τ . Without loss of generality, suppose σp C τ . Since τ C σq,
therefore σp C σq: a contradiction.
7
Monomials and polynomials
The purpose of this section is to introduce definitions and properties about terms. These definitions and properties are purely Boolean. Let k, m, n be nonnegative integers. An m-vector is a map s associating with every positive integer i ≤ m an element s(i) of {0, 1}. A (k, m, n)-correspondence is a map f associating with every m-vector s a function fs : {0, 1}k −→ {0, 1}n. Let f be a (k, m,
n)-correspondence. An n-monomial is a term of the form – xβ1
where (β1, . . . , βn) ∈ {0, 1}n. For all m-vectors s, considering a term a in
TERm,n as a formula in Classical Propositional Logic (CPL), we define:
– mons(n, a) = {xβ11∩ . . . ∩ xβnn : a is a tautological consequence of p s(1) 1 ∩
. . . ∩ ps(m)m ∩ xβ11∩ . . . ∩ xβnn}.
Hence, for all m-vectors s, considering a as a formula in CPL, mons(n, a) is
nothing but the set of all conjunctions of the form xβ1
1 ∩ . . . ∩ xβnn that appear
in the Disjunctive Normal Form of a in conjunction with the conjunction ps(1)1 ∩ . . . ∩ ps(m)m . An n-polynomial is a term of the form
– (xβ11 1 ∩ . . . ∩ x β1n n ) ∪ . . . ∪ (x βj1 1 ∩ . . . ∩ x βjn n )
where j is a nonnegative integer and (β11, . . . , β1n), . . . , (βj1, . . . , βjn) ∈ {0, 1}n.
Remark that for all m-vectors s and for all terms a in TERm,n,S mons(n, a)
is an n-polynomial.
Proposition 4. Let a ∈ TERm,n. Considered as formulas in CPL, the
follow-ing terms are equivalent: – a,
– S{ps(1)1 ∩ . . . ∩ ps(m)m ∩S mons(n, a) : s is an m-vector}.
Proof. By the definition of mons, s being an arbitrary m-vector.
For all positive integers i ≤ n, let πi : {0, 1}n −→ {0, 1} be the function such
that for all (β1, . . . , βn) ∈ {0, 1}n,
– πi(β1, . . . , βn) = βi.
For all m-vectors s and for all (β1, . . . , βn) ∈ {0, 1}n, we define:
– fs−1(β1, . . . , βn) = {(α1, . . . , αk) ∈ {0, 1}k : fs(α1, . . . , αk) = (β1, . . . , βn)}.
Obviously, for all m-vectors s and for all (β1, . . . , βn) ∈ {0, 1}n, fs−1(β1, . . . , βn)
⊆ {0, 1}k. For all m-vectors s and for all positive integers i ≤ n, we define:
– ∆s,i= {(α1, . . . , αk) ∈ {0, 1}k: πi(fs(α1, . . . , αk)) = 1},
– cs,i=S{xα11∩ . . . ∩ x αk
k : (α1, . . . , αk) ∈ ∆s,i}.
Obviously, for all positive integers i ≤ n, ∆s,i ⊆ {0, 1}k and cs,i is a
k-polyno-mial. Remark that ∆s,iand cs,i depend on f — more precisely, on fs — too.
Proposition 5. For all m-vectors s and for all (β1, . . . , βn) ∈ {0, 1}n,
consid-ered as formulas in CPL, the following terms are equivalent: – S{xα1 1 ∩ . . . ∩ x αk k : (α1, . . . , αk) ∈ f −1 s (β1, . . . , βn)}, – cβ1 s,1∩ . . . ∩ cβs,nn.
Proof. Let s be an m-vector and (β1, . . . , βn) ∈ {0, 1}n. It suffices to show that
considered as formulas in CPL, for all θ1, . . . , θk ∈ {0, 1}, if x1, . . . , xk are
re-spectively interpreted by θ1, . . . , θk then S{xα11 ∩ . . . ∩ x αk
k : (α1, . . . , αk) ∈
fs−1(β1, . . . , βn)} is equivalent to 1 iff cβs,11 ∩ . . . ∩ c βn
s,n is equivalent to 1. Let
θ1, . . . , θk ∈ {0, 1}. Let x1, . . . , xk be respectively interpreted by θ1, . . . , θk.
Suppose S{xα1
1 ∩ . . . ∩ x αk
k : (α1, . . . , αk) ∈ fs−1(β1, . . . , βn)} is equivalent to
1. Hence, (θ1, . . . , θk) ∈ fs−1(β1, . . . , βn). Thus, fs(θ1, . . . , θk) = (β1, . . . , βn).
For the sake of the contradiction, suppose cβ1
s,1∩ . . . ∩ cβs,nn is equivalent to 0.
Let i ≤ n be a positive integer such that cβi
s,i is equivalent to 0. Since either
βi = 0, or βi = 1, therefore we have to consider two cases. In the former case,
βi = 0 and therefore S{xα11 ∩ . . . ∩ x αk
k : (α1, . . . , αk) ∈ ∆s,i} is equivalent
to 1. Consequently, (θ1, . . . , θk) ∈ ∆s,i. Hence, πi(fs(θ1, . . . , θk)) = 1. Since
fs(θ1, . . . , θk) = (β1, . . . , βn), therefore βi = 1: a contradiction. In the
lat-ter case, βi = 1 and therefore S{xα11 ∩ . . . ∩ x αk
k : (α1, . . . , αk) ∈ ∆s,i} is
equivalent to 0. Thus, (θ1, . . . , θk) 6∈ ∆s,i. Hence, πi(fs(θ1, . . . , θk)) = 0. Since
fs(θ1, . . . , θk) = (β1, . . . , βn), therefore βi= 0: a contradiction.
Suppose cβ1
s,1∩ . . . ∩ cβs,nn is equivalent to 1. Let i ≤ n be an arbitrary
posi-tive integer. Since cβ1
s,1∩ . . . ∩ cβs,nn is equivalent to 1, therefore c βi
s,i is
equiv-alent to 1. Since either βi = 0, or βi = 1, therefore we have to consider
two cases. In the former case, βi = 0 and therefore cs,i is equivalent to 0.
Hence, (θ1, . . . , θk) 6∈ ∆s,i. Thus, πi(fs(θ1, . . . , θk)) = 0. Since βi = 0, therefore
πi(fs(θ1, . . . , θk)) = βi. In the latter case, βi = 1 and therefore cs,i is
equiva-lent to 1. Consequently, (θ1, . . . , θk) ∈ ∆s,i. Hence, πi(fs(θ1, . . . , θk)) = 1. Since
βi= 1, therefore πi(fs(θ1, . . . , θk)) = βi. In both cases, πi(fs(θ1, . . . , θk)) = βi.
Since i was arbitrary, therefore fs(θ1, . . . , θk) = (β1, . . . , βn). Thus, (θ1, . . . , θk) ∈
fs−1(β1, . . . , βn). Consequently,S{xα11 ∩ . . . ∩ x αk
k : (α1, . . . , αk) ∈ fs−1(β1, . . . ,
βn)} is equivalent to 1.
The property about m-vector s and (β1, . . . , βn) ∈ {0, 1}ndescribed in
Proposi-tion 5 is purely Boolean. It will be used in the proof of ProposiProposi-tion 6 which will itself be used in the proof of Proposition 12, our main result.
8
Equivalence relations between tuples of bits
Let k, m, n be nonnegative integers. Let (a1, . . . , an) ∈ TERnm,k. For all
m-vectors s, we define on {0, 1}k the equivalence relation ∼k,s
(a1,...,an) as follows: – (α1, . . . , αk) ∼
k,s
(a1,...,an) (α
0
1, . . . , α0k) if for all positive integers i ≤ n, x α1 1 ∩ . . . ∩ xαk k ∈ mons(k, ai) iff x α01 1 ∩ . . . ∩ x α0k k ∈ mons(k, ai).
Obviously, for all m-vectors s, ∼k,s(a
1,...,an) has at most 2
n equivalence classes on
for all (α1, . . . , αk), (α01, . . . , α0k) ∈ {0, 1} k, if f s(α1, . . . , αk) = fs(α01, . . . , α0k) then (α1, . . . , αk) ∼ k,s (a1,...,an)(α 0
1, . . . , α0k). For all m-vectors s, by means of f —
more precisely, of fs —, we define the n-tuple (bs,1, . . . , bs,n) of n-polynomials
as follows: – bs,i=S{xβ11∩ . . . ∩ x βn n : x α1 1 ∩ . . . ∩ x αk k ∈ mons(k, ai) and fs(α1, . . . , αk) = (β1, . . . , βn)}.
An n-tuple (b1, . . . , bn) ∈ TERnm,nof terms is properly obtained from (a1, . . . , an)
if for all positive integers i ≤ n, considered as formulas in CPL, the following terms are equivalent:
– bi,
– S{ps(1)1 ∩ . . . ∩ p s(m)
m ∩ bs,i: s is an m-vector}.
For all m-vectors s and for all (β1, . . . , βn) ∈ {0, 1}n, let fs−1(β1, . . . , βn) be as
in Section 7. For all m-vectors s and for all positive integers i ≤ n, let ∆s,i and
cs,ibe as in Section 7. A substitution υ is properly obtained from (a1, . . . , an) if
for all variables y, if y 6∈ {x1, . . . , xn} then υ(y) = y and for all positive integers
i ≤ n, considered as formulas in CPL, the following terms are equivalent: – υ(xi),
– S{ps(1)1 ∩ . . . ∩ ps(m)m ∩ cs,i: s is an m-vector}.
The proof of the following result uses the purely Boolean property described in Proposition 5.
Proposition 6. Let (a1, . . . , an) ∈ TERnm,k, (b1, . . . , bn) ∈ TERnm,n be an
n-tuple of terms and υ be a substitution. If (b1, . . . , bn) and υ are properly obtained
from (a1, . . . , an) then for all positive integers i ≤ n, considered as formulas in
CPL, the following terms are equivalent: – ai,
– ¯υ(bi).
Proof. Suppose (b1, . . . , bn) and υ are properly obtained from (a1, . . . , an). Let
i ≤ n be a positive integer. Considered as formulas in CPL, the following terms are equivalent: 1. ¯υ(bi). 2. S{ps(1)1 ∩ . . . ∩ p s(m) m ∩ ¯υ(bs,i) : s is an m-vector}. 3. S{ps(1)1 ∩. . .∩ps(m)m ∩υ(x1)β1∩. . .∩υ(xn)βn: s is an m-vector, xα11∩. . .∩x αk k ∈ mons(k, ai) and fs(α1, . . . , αk) = (β1, . . . , βn)}. 4. S{ps(1)1 ∩ . . . ∩ p s(m) m ∩ cβs,11 ∩ . . . ∩ cβs,nn : s is an m-vector, x α1 1 ∩ . . . ∩ x αk k ∈ mons(k, ai) and fs(α1, . . . , αk) = (β1, . . . , βn)}. 5. S{ps(1)1 ∩ . . . ∩ p s(m) m ∩ x α01 1 ∩ . . . ∩ x α0k k : s is an m-vector, x α1 1 ∩ . . . ∩ x αk k ∈ mons(k, ai), fs(α1, . . . , αk) = (β1, . . . , βn) and (α01, . . . , α0k) ∈ fs−1(β1, . . . , βn)}.
6. S{ps(1)1 ∩ . . . ∩ p s(m)
m ∩S mons(k, ai) : s is an m-vector}.
7. ai.
The equivalence between 1 and 2 is a consequence of the definition of (b1, . . . , bn).
The equivalence between 2 and 3 is a consequence of the definition of (bs,1, . . . ,
bs,n), s being an arbitrary m-vector. The equivalence between 3 and 4 is a
conse-quence of the definition of υ. The equivalence between 4 and 5 is a conseconse-quence of Proposition 5. The equivalence between 5 and 6 is a consequence of the defini-tions of monsand ∼k,s(a
1,...,an)and the fact that for all (α1, . . . , αk), (α
0 1, . . . , α0k) ∈ {0, 1}k, if f s(α1, . . . , αk) = fs(α01, . . . , α0k) then (α1, . . . , αk) ∼k,s(a 1,...,an)(α 0 1, . . . ,
α0k), s being an arbitrary m-vector. The equivalence between 6 and 7 is a con-sequence of Proposition 4.
Proposition 7. Let (a1, . . . , an) ∈ TERnm,k and (b1, . . . , bn) ∈ TERnm,n be an
n-tuple of terms. Let W be a nonempty set. If (b1, . . . , bn) is properly obtained
from (a1, . . . , an) then:
– for all valuations V on W , there exists a valuation V0 on W such that for all positive integers i ≤ n, ¯V(ai) = ¯V0(bi),
– for all valuations V on W , there exists a valuation V0 on W such that for
all positive integers i ≤ n, ¯V(bi) = ¯V0(ai).
Proof. Let V be a valuation on W . Let V0 be a valuation on W such that for
all positive integers i ≤ m, V0(pi) = V(pi) and for all positive integers i ≤ n,
V0(x
i) = ¯V(S{p s(1)
1 ∩ . . . ∩ p s(m)
m ∩ cs,i : s is an m-vector}). For all positive
integers i ≤ n, the following subsets of W are equal: 1. ¯V0(b i). 2. S{ ¯V0(ps(1) 1 ∩ . . . ∩ p s(m) m ) ∩ ¯V0(bs,i) : s is an m-vector}. 3. S{ ¯V0(ps(1) 1 ∩. . .∩p s(m) m )∩ ¯V0(xβ11∩. . .∩x βn n ) : s is an m-vector, x α1 1 ∩. . .∩x αk k ∈ mons(k, ai) and fs(α1, . . . , αk) = (β1, . . . , βn)}. 4. ¯V(S{ps(1)1 ∩ . . . ∩ ps(m)m ∩ cβs,11 ∩ . . . ∩ cβs,nn : s is an m-vector, x α1 1 ∩ . . . ∩ x αk k ∈ mons(k, ai) and fs(α1, . . . , αk) = (β1, . . . , βn)}). 5. ¯V(S{ps(1)1 ∩ . . . ∩ p s(m) m ∩ x α01 1 ∩ . . . ∩ x α0k k : s is an m-vector, x α1 1 ∩ . . . ∩ x αk k ∈ mons(k, ai), fs(α1, . . . , αk) = (β1, . . . , βn) and (α01, . . . , α0k) ∈ f −1 s (β1, . . . , βn)}). 6. ¯V(S{ps(1)1 ∩ . . . ∩ p s(m) m ∩S mons(k, ai) : s is an m-vector}). 7. ¯V(ai).
The equivalence between 1 and 2 is a consequence of the definition of (b1, . . . , bn).
The equivalence between 2 and 3 is a consequence of the definition of (bs,1, . . . ,
bs,n), s being an arbitrary m-vector. The equivalence between 3 and 4 is a
conse-quence of the definition of V0. The equivalence between 4 and 5 is a consequence of Proposition 5. The equivalence between 5 and 6 is a consequence of the defini-tions of monsand ∼k,s(a
1,...,an)and the fact that for all (α1, . . . , αk), (α
0 1, . . . , α0k) ∈ {0, 1}k, if f s(α1, . . . , αk) = fs(α01, . . . , α0k) then (α1, . . . , αk) ∼k,s(a 1,...,an)(α 0 1, . . . ,
α0k), s being an arbitrary m-vector. The equivalence between 6 and 7 is a con-sequence of Proposition 4.
Let V be a valuation on W . Let V0 be a valuation on W such that for all positive integers i ≤ m, V0(pi) = V(pi) and for all m-vectors s and for all (α1, . . . , αk) ∈
{0, 1}k, ¯V0(ps(1) 1 ∩. . .∩p s(m) m ∩xα11∩. . .∩x αk k ) = ¯V(p s(1) 1 ∩. . .∩p s(m) m ∩xβ11∩. . .∩xβnn)
where (β1, . . . , βn) ∈ {0, 1}n is such that fs(α1, . . . , αk) = (β1, . . . , βn). For all
positive integers i ≤ n, the following subsets of W are equal: 1. ¯V0(a i). 2. S{ ¯V0(ps(1) 1 ∩ . . . ∩ p s(m) m ) ∩ ¯V0(S mons(k, ai)) : s is an m-vector}. 3. ¯V(S{ps(1)1 ∩ . . . ∩ ps(m)m ∩ xβ11∩ . . . ∩ xβnn: s is an m-vector, x α1 1 ∩ . . . ∩ x αk k ∈ mons(k, ai) and fs(α1, . . . , αk) = (β1, . . . , βn)}). 4. ¯V(S{ps(1)1 ∩ . . . ∩ p s(m) m ∩ bs,i: s is an m-vector}). 5. ¯V(bi).
The equivalence between 1 and 2 is a consequence of Proposition 4. The equiv-alence between 2 and 3 is a consequence of the definitions of mons and V0, s
being an arbitrary m-vector. The equivalence between 3 and 4 is a consequence of the definition of (bs,1, . . . , bs,n), s being an arbitrary m-vector. The equivalence
between 4 and 5 is a consequence of the definition of (b1, . . . , bn).
9
Equivalence relations between formulas
Let k, m, n be nonnegative integers and C be a class of frames. We define on FORm,n the equivalence relation ≡
m,n
C as follows:
– ϕ ≡m,nC ψ if ϕ ↔ ψ is C-valid.
Since each formula ϕ in FORm,n is a combination of formulas of the form
C(a, b), a ≡ b and con(a) where a and b are terms in TERm,n, therefore
≡m,nC has finitely many equivalence classes on FORm,n. Let An be the set of
all n-tuples of terms. Note that n-tuples of terms in An may contain
occur-rences of constants outside {p1, . . . , pm} and occurrences of variables outside
{x1, . . . , xn}. Given a model (W, R, V) on a frame in C and (a1, . . . , an) ∈ An,
let Φ(W,R,V)(a
1,...,an)be the set of all equational formulas ϕ(x1, . . . , xn) in FORm,nsuch that (W, R, V) |= ϕ(a1, . . . , an). Consider a complete list of representatives for
each equivalence class on Φ(W,R,V)(a
1,...,an)modulo ≡
m,n
C and let ϕ (W,R,V)
(a1,...,an)(x1, . . . , xn) be their conjunction. We define on An the equivalence relation ∼=
m,n
C as follows:
– (a1, . . . , an) ∼= m,n
C (b1, . . . , bn) if for all formulas ϕ(x1, . . . , xn) in FORm,n,
ϕ(a1, . . . , an) is C-valid iff ϕ(b1, . . . , bn) is C-valid.
Since ≡m,nC has finitely many equivalence classes on FORm,n, therefore ∼= m,n C has
finitely many equivalence classes on An. Now, we define on An the equivalence
– (a1, . . . , an) ' m,n
C (b1, . . . , bn) if for all equational formulas ϕ(x1, . . . , xn) in
FORm,n, ϕ(a1, . . . , an) is C-valid iff ϕ(b1, . . . , bn) is C-valid.
Obviously, ∼=m,nC is finer than 'm,nC . Since ∼=m,nC has finitely many equivalence classes on An, therefore 'm,nC has finitely many equivalence classes on An.
Proposition 8. Let (a1, . . . , an) ∈ TERnm,k and (b1, . . . , bn) ∈ TERnm,n be
an n-tuple of terms. If (b1, . . . , bn) is properly obtained from (a1, . . . , an) then
(a1, . . . , an) ' m,n
C (b1, . . . , bn).
Proof. By Proposition 7.
Proposition 9. TERnm,nconstitutes a complete set of representatives for each equivalence class on An modulo 'm,nC .
Proof. By Proposition 8.
Proposition 10. Let (W, R, V) be a model on a frame in C and (a1, . . . , an) ∈
An. (W, R, V) |= ϕ (W,R,V)
(a1,...,an)(a1, . . . , an). Proof. By the definition of ϕ(W,R,V)(a
1,...,an)(x1, . . . , xn).
Proposition 11. If C is balanced then for all (a1, . . . , an), (b1, . . . , bn) ∈ An, if
(a1, . . . , an) ' m,n
C (b1, . . . , bn) then (a1, . . . , an) ∼= m,n
C (b1, . . . , bn).
Proof. Suppose C is balanced. Let (a1, . . . , an), (b1, . . . , bn) ∈ An be such that
(a1, . . . , an) 'm,nC (b1, . . . , bn) and (a1, . . . , an) 6∼=m,nC (b1, . . . , bn). Let ϕ(x1, . . . ,
xn) be a formula in FORm,nsuch that ϕ(a1, . . . , an) is C-valid not-iff ϕ(b1, . . . ,
bn) is C-valid. Without loss of generality, let us assume that ϕ(a1, . . . , an) is
C-valid and ϕ(b1, . . . , bn) is not C-valid. Remind that ≡ may not be the only
pred-icate occurring in ϕ(x1, . . . , xn). Since C is balanced, therefore let (W, R, V) be a
balanced model on a countable frame in C such that (W, R, V) 6|= ϕ(b1, . . . , bn).
By Proposition 10, (W, R, V) |= ϕ(W,R,V)(b
1,...,bn)(b1, . . . , bn). Hence, ¬ϕ
(W,R,V)
(b1,...,bn)(b1, . . . , bn) is not C-valid. Remind that ϕ
(W,R,V)
(b1,...,bn)(x1, . . . , xn) is equational. Since (a1, . . . , an) 'm,nC (b1, . . . , bn), therefore ¬ϕ
(W,R,V)
(b1,...,bn)(a1, . . . , an) is not C-valid. Since C is balanced, therefore let (W0, R0, V0) be a balanced model on a countable frame in C such that (W0, R0, V0) |= ϕ(W,R,V) (b1,...,bn)(a1, . . . , an). Now, consider (β1, . . . , βn) ∈ {0, 1}n. If ¯V(bβ1 1 ∩ . . . ∩ b βn n ) = ∅ then (W, R, V) |= b β1 1 ∩ . . . ∩ b βn n ≡ 0. Thus, ϕ(W,R,V)(b 1,...,bn)(x1, . . . , xn) → x β1 1 ∩ . . . ∩ xβnn ≡ 0 is C-valid. Since (W0, R0, V0) |= ϕ(W,R,V)(b 1,...,bn)(a1, . . . , an), therefore (W 0, R0, V0) |= aβ1 1 ∩ . . . ∩ a βn n ≡ 0. Conse-quently, ¯V0(aβ1
1 ∩ . . . ∩ aβnn) = ∅. Similarly, the reader may easily verify that
if ¯V(bβ1 1 ∩ . . . ∩ bβnn) = W then ¯V0(a β1 1 ∩ . . . ∩ aβnn) = W0and if ¯V(b β1 1 ∩ . . . ∩ bβnn) is
infinite and coinfinite then ¯V0(aβ1
1 ∩. . .∩aβnn) is infinite and coinfinite. In all cases,
there exists a bijection g(β1,...,βn)from ¯V(b
β1
1 ∩. . .∩bβnn) to ¯V0(a β1
1 ∩. . .∩aβnn). Let
g be the union of all g(β1,...,βn)when (β1, . . . , βn) describes {0, 1}
may easily verify that g is a bijection from W to W0 such that for all u ∈ W and for all (β1, . . . , βn) ∈ {0, 1}n, u ∈ ¯V(b β1 1 ∩ . . . ∩ bβnn) iff g(u) ∈ ¯V0(a β1 1 ∩ . . . ∩ aβnn).
Let R0g be the binary relation on W0 defined by u0Rg0v0 if g−1(u0)Rg−1(v0). Ob-viously, g is an isomorphism from (W, R) to (W0, R0g). Since ϕ(a1, . . . , an) is
C-valid, therefore (W0, R0g, V0) |= ϕ(a1, . . . , an). Hence, (W, R, V) |= ϕ(b1, . . . , bn):
a contradiction.
10
Unification type
Let C be a class of frames.
Proposition 12. If C is balanced then unification in C is either finitary, or unitary. Moreover, if C agrees with unions then unification in C is finitary. Proof. Suppose C is balanced. Let ϕ be an arbitrary C-unifiable formula. Let σ be an arbitrary substitution such that ¯σ(ϕ) is C-valid. Let m, n ≥ 0 be such that ϕ’s constants form a subset of {p1, . . . , pm} and ϕ’s variables form
a subset of {x1, . . . , xn}. Let σ0 be the substitution defined by σ0(xi) = σ(xi)
for all i = 1 . . . n and σ0(y) = y for all y not in {x1, . . . , xn}. Obviously,
σ0 is a unifier of ϕ too. Now, it may happen that for some i ∈ {1, . . . , n}, σ0(xi) contains extra constants outside {p1, . . . , pm}. If it is, then let q1, . . . , ql
be the list of these extra constants. Take new variables z1, . . . , zl and define
σ00 by uniformly replacing in σ0(x1), . . . , σ0(xn) each occurrence of q1, . . . , ql
by, respectively, z1, . . . , zl. Obviously, σ00 is a unifier of ϕ too. As a result,
for all constants q, if q 6∈ {p1, . . . , pm} then for all positive integers i ≤ n,
q does not occur in σ00(x
i) and for all variables y, if y 6∈ {x1, . . . , xn} then
σ00(y) = y. Let k be a nonnegative integer and (a1, . . . , an) ∈ TERnm,kbe such
that for all positive integers i ≤ n, σ00(xi) = ai. Since ¯σ00(ϕ) is C-valid,
there-fore ϕ(a1, . . . , an) is C-valid. For all m-vectors s, let ∼k,s(a
1,...,an) be as in Sec-tion 8. Let f be a (k, m, n)-correspondence such that for all m-vectors s and for all (α1, . . . , αk), (α01, . . . , α0k) ∈ {0, 1} k, if f s(α1, . . . , αk) = fs(α01, . . . , α0k) then (α1, . . . , αk) ∼ k,s (a1,...,an)(α 0
1, . . . , α0k). For all m-vectors s and for all (β1, . . . , βn) ∈
{0, 1}n, let f−1
s (β1, . . . , βn) be as in Section 7. For all m-vectors s and for all
posi-tive integers i ≤ n, let ∆s,iand cs,ibe as in Section 7. Let (b1, . . . , bn) ∈ TERnm,n
be an n-tuple of terms properly obtained from (a1, . . . , an). By Proposition 8,
(a1, . . . , an) 'm,nC (b1, . . . , bn). Since C is balanced, therefore by Proposition 11,
(a1, . . . , an) ∼= m,n
C (b1, . . . , bn). Let τ be the substitution such that for all variables
y, if y 6∈ {x1, . . . , xn} then τ (y) = y and for all positive integers i ≤ n, τ (xi) = bi.
Remark that (τ (x1), . . . , τ (xn)) ∈ TERnm,n. Moreover, since ϕ(a1, . . . , an) is
C-valid and (a1, . . . , an) ∼= m,n
C (b1, . . . , bn), therefore ϕ(b1, . . . , bn) is C-valid. Hence,
τ is a C-unifier of ϕ. Let υ be a substitution properly obtained from (a1, . . . , an).
By Proposition 6, for all positive integers i ≤ n, considered as formulas in CPL, the terms ai and ¯υ(bi) are equivalent. Thus, for all positive integers i ≤ n,
¯
υ(τ (xi)) ≡ σ00(xi) is C-valid. Consequently, τ ◦ υ 'C σ00. Hence, τ C σ00. By
(τ (x1), . . . , τ (xn)) ∈ TERnm,n, therefore ϕ is either C-finitary, or C-unitary. Since
ϕ was arbitrary, therefore unification in C is either finitary, or unitary. Now, sup-pose C agrees with unions. By the examples considered in Section 6, unification in C is not unitary. Since unification in C is either finitary, or unitary, therefore unification in C is finitary.
It follows from the above discussion that unification in Call, Cind and Ccon is
finitary.
11
Conclusion
We anticipate a number of further investigations. For example, about the com-putability of the unification problem in Contact Logic. It is proved in [5] that with respect to most classes of frames, in Contact Logic, when one restricts the discussion to con-free formulas, elementary unification is N P -complete whereas unification with constants is decidable, its exact complexity being still unknown. By means of Propositions 9 and 11, one can prove that with respect to balanced classes of frames, elementary unification and unification with constants are de-cidable in Contact Logic when one does not restrict the discussion to con-free formulas, but its exact complexity is unknown. In this respect, we believe that arguments developed in [1] could be used.
Acknowledgements
Special acknowledgement is heartily granted to Tinko Tinchev (Sofia University, Bulgaria) for his valuable remarks. We also make a point of thanking the referees for their feedback: their helpful comments and their useful suggestions have been essential for improving the correctness and the readability of a preliminary version of this paper.
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